Subgroups — C₆₇×#34379 #30713630

Order 17152.

3082 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2377	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2361	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2345	C₂	1	2	yes	maximal, complemented
5	D₄×C₂²⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2325	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2324	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2315	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2312	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₄	1	16	yes	complemented
17	C₂×#121×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×#121×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
22	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
23	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#390×C₆₇	C₂²	1	4	yes	complemented
30	#390×C₆₇	C₂²	1	8	yes	complemented
31	#389×C₆₇	C₂²	1	4	yes	complemented
32	#388×C₆₇	C₂²	1	8	yes	complemented
33	#388×C₆₇	C₂²	1	8	yes	complemented
34	#387×C₆₇	C₂²	1	8	yes	complemented
35	#386×C₆₇	C₂²	1	8	yes	complemented
36	#386×C₆₇	C₂²	1	8	yes	complemented
37	#386×C₆₇	C₂²	1	8	yes	complemented
38	#385×C₆₇	C₂²	1	8	yes	complemented
39	#385×C₆₇	C₂²	1	8	yes	complemented
40	#381×C₆₇	C₂²	1	2	yes	complemented
41	#380×C₆₇	C₂²	1	2	yes	—
42	#380×C₆₇	C₂²	1	2	yes	—
43	#380×C₆₇	C₂²	1	1	yes	characteristic
44	#380×C₆₇	C₂²	1	1	yes	characteristic
45	#380×C₆₇	C₂²	1	4	yes	—
46	#380×C₆₇	C₂²	1	2	yes	complemented
47	#380×C₆₇	C₂²	1	2	yes	complemented
48	#380×C₆₇	C₂²	1	2	yes	complemented
49	#379×C₆₇	C₂²	1	2	yes	—
50	#379×C₆₇	C₂²	1	4	yes	—
51	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
52	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
53	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
55	C₂×#95×C₆₇	C₂²	1	2	yes	complemented
56	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
57	C₂⁵×C₆₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₇	—	4	4	no	—
59	C₂²×D₄×C₆₇	—	4	8	no	—
60	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂²×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
62	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
66	C₂³×C₄×C₆₇	—	4	4	no	—
67	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	complemented
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	#121×C₆₇	C₂×C₄	1	16	yes	complemented
78	#121×C₆₇	C₂×C₄	1	16	yes	complemented
79	#121×C₆₇	C₂×C₄	1	16	yes	complemented
80	#121×C₆₇	C₂×C₄	1	16	yes	complemented
81	C₄×D₄×C₆₇	—	4	8	no	—
82	C₄×D₄×C₆₇	—	4	16	no	—
83	#117×C₆₇	—	4	4	no	—
84	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
109	#95×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	#95×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	8	yes	—
113	#95×C₆₇	C₂³	1	2	yes	—
114	#95×C₆₇	C₂³	1	2	yes	—
115	#95×C₆₇	C₂³	1	8	yes	—
116	C₂⁴×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	—	2	4	no	—
118	C₂⁴×C₆₇	—	2	2	no	—
119	C₂⁴×C₆₇	—	2	4	no	—
120	C₂⁴×C₆₇	—	2	4	no	—
121	C₂⁴×C₆₇	—	4	2	no	—
122	C₂⁴×C₆₇	—	4	4	no	—
123	C₂⁴×C₆₇	—	4	4	no	—
124	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
126	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂×D₄×C₆₇	—	2	16	no	—
131	C₂×D₄×C₆₇	—	2	8	no	—
132	C₂×D₄×C₆₇	—	4	16	no	—
133	C₂×D₄×C₆₇	—	4	8	no	—
134	C₂×D₄×C₆₇	—	4	4	no	—
135	C₂×D₄×C₆₇	—	4	32	no	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
137	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
138	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
139	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	8	no	—
144	C₂²×C₄×C₆₇	—	4	16	no	—
145	C₂²×C₄×C₆₇	—	4	2	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	—	4	4	no	—
150	C₂²×C₄×C₆₇	—	4	8	no	—
151	C₂²×C₄×C₆₇	—	4	4	no	—
152	C₂²×C₄×C₆₇	—	4	8	no	—
153	C₂²×C₄×C₆₇	—	4	8	no	—
154	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	4	4	no	—
171	C₄⋊C₄×C₆₇	—	4	2	no	—
172	C₄⋊C₄×C₆₇	—	4	2	no	—
173	C₄⋊C₄×C₆₇	—	4	8	no	—
174	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
176	C₂²⋊C₄×C₆₇	—	2	8	no	—
177	C₂²⋊C₄×C₆₇	—	2	4	no	—
178	C₂²⋊C₄×C₆₇	—	2	8	no	—
179	C₂²⋊C₄×C₆₇	—	2	4	no	—
180	C₂²⋊C₄×C₆₇	—	4	4	no	—
181	C₂²⋊C₄×C₆₇	—	4	8	no	—
182	C₂²⋊C₄×C₆₇	—	4	4	no	—
183	C₂²⋊C₄×C₆₇	—	4	8	no	—
184	C₂²⋊C₄×C₆₇	—	4	8	no	—
185	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
186	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
187	C₄²×C₆₇	—	2	8	no	—
188	C₄²×C₆₇	—	4	8	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	1	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	8	no	—
195	C₂³×C₆₇	—	2	8	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	2	8	no	—
200	C₂³×C₆₇	—	2	8	no	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	1	no	—
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	2	16	no	—
205	C₂³×C₆₇	—	4	4	no	—
206	C₂³×C₆₇	—	4	8	no	—
207	C₂³×C₆₇	—	4	8	no	—
208	C₂³×C₆₇	—	4	8	no	—
209	C₂³×C₆₇	—	4	8	no	—
210	C₂³×C₆₇	—	8	8	no	—
211	C₂³×C₆₇	—	8	8	no	—
212	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
213	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
214	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
215	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
216	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
217	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
218	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
219	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
220	D₄×C₆₇	—	2	32	no	—
221	D₄×C₆₇	—	2	16	no	—
222	D₄×C₆₇	—	4	8	no	—
223	D₄×C₆₇	—	4	16	no	—
224	C₂×C₄×C₆₇	—	2	8	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	8	no	—
229	C₂×C₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	2	8	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	8	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	8	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	4	4	no	—
241	C₂×C₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	—	4	4	no	—
243	C₂×C₄×C₆₇	—	4	8	no	—
244	C₂×C₄×C₆₇	—	8	8	no	—
245	C₂×C₄×C₆₇	—	8	16	no	—
246	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
247	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
248	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
249	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
250	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
251	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	4	no	—
257	C₂²×C₆₇	—	2	4	no	—
258	C₂²×C₆₇	—	2	4	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	8	no	—
262	C₂²×C₆₇	—	2	8	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	8	no	—
266	C₂²×C₆₇	—	4	16	no	—
267	C₂²×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	4	8	no	—
269	C₂²×C₆₇	—	4	8	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	8	16	no	—
272	C₂²×C₆₇	#518	1	4	yes	—
273	C₂²×C₆₇	#520	1	4	yes	—
274	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
275	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
276	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
277	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	2	no	—
282	C₄×C₆₇	—	2	8	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	8	no	—
286	C₄×C₆₇	#518	1	4	yes	—
287	#34379	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₆₇	—	2	4	no	—
289	C₂×C₆₇	—	2	4	no	—
290	C₂×C₆₇	—	4	2	no	—
291	C₂×C₆₇	—	4	8	no	—
292	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
293	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
294	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
295	C₂×C₆₇	#3536	1	2	yes	—
296	C₂×C₆₇	#3537	1	2	yes	—
297	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
298	#2377	C₂×C₆₇	1	4	yes	complemented
299	#2361	C₂×C₆₇	1	4	yes	complemented
300	#2345	C₂×C₆₇	1	2	yes	complemented
301	D₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
302	D₄×C₂²⋊C₄	C₂×C₆₇	1	4	yes	complemented
303	#2325	C₂×C₆₇	1	4	yes	complemented
304	#2324	C₂×C₆₇	1	4	yes	complemented
305	#2315	C₂×C₆₇	1	2	yes	complemented
306	#2312	C₂×C₆₇	1	2	yes	complemented
307	#2280	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₆₇	#34379	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
311	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
312	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#121	C₄×C₆₇	1	16	yes	complemented
314	C₂×#121	C₂²×C₆₇	1	2	yes	—
315	C₂×#121	C₂²×C₆₇	1	2	yes	—
316	C₂×#121	C₂²×C₆₇	1	2	yes	—
317	C₂×#121	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
319	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
320	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
321	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
323	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
324	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
325	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
326	#390	C₂²×C₆₇	1	4	yes	complemented
327	#390	C₂²×C₆₇	1	8	yes	complemented
328	#389	C₂²×C₆₇	1	4	yes	complemented
329	#388	C₂²×C₆₇	1	8	yes	complemented
330	#388	C₂²×C₆₇	1	8	yes	complemented
331	#387	C₂²×C₆₇	1	8	yes	complemented
332	#386	C₂²×C₆₇	1	8	yes	complemented
333	#386	C₂²×C₆₇	1	8	yes	complemented
334	#386	C₂²×C₆₇	1	8	yes	complemented
335	#385	C₂²×C₆₇	1	8	yes	complemented
336	#385	C₂²×C₆₇	1	8	yes	complemented
337	#381	C₂²×C₆₇	1	2	yes	complemented
338	#380	C₂²×C₆₇	1	1	yes	characteristic
339	#380	C₂²×C₆₇	1	2	yes	—
340	#380	C₂²×C₆₇	1	1	yes	characteristic
341	#380	C₂²×C₆₇	1	4	yes	—
342	#380	C₂²×C₆₇	1	2	yes	—
343	#380	C₂²×C₆₇	1	2	yes	complemented
344	#380	C₂²×C₆₇	1	2	yes	complemented
345	#380	C₂²×C₆₇	1	2	yes	complemented
346	#379	C₂²×C₆₇	1	4	yes	—
347	#379	C₂²×C₆₇	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
349	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
350	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
351	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
352	C₂×#95	C₂²×C₆₇	1	2	yes	complemented
353	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
354	C₂⁵	C₂³×C₆₇	1	2	yes	—
355	C₂²×D₄	—	4	4	no	—
356	C₂²×D₄	—	4	8	no	—
357	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
358	C₂²×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
359	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
360	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
361	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
362	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
363	C₂³×C₄	—	4	4	no	—
364	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	complemented
365	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
369	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
370	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
372	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
374	#121	C₂×C₄×C₆₇	1	16	yes	complemented
375	#121	C₂×C₄×C₆₇	1	16	yes	complemented
376	#121	C₂×C₄×C₆₇	1	16	yes	complemented
377	#121	C₂×C₄×C₆₇	1	16	yes	complemented
378	C₄×D₄	—	4	8	no	—
379	C₄×D₄	—	4	16	no	—
380	#117	—	4	4	no	—
381	C₂×C₄⋊C₄	—	4	4	no	—
382	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
390	C₂×C₂²⋊C₄	—	4	4	no	—
391	C₂×C₂²⋊C₄	—	4	8	no	—
392	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
393	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
405	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
406	#95	C₂³×C₆₇	1	4	yes	—
407	#95	C₂³×C₆₇	1	4	yes	—
408	#95	C₂³×C₆₇	1	4	yes	—
409	#95	C₂³×C₆₇	1	8	yes	—
410	#95	C₂³×C₆₇	1	2	yes	—
411	#95	C₂³×C₆₇	1	2	yes	—
412	#95	C₂³×C₆₇	1	8	yes	—
413	C₂⁴	—	2	2	no	—
414	C₂⁴	—	2	4	no	—
415	C₂⁴	—	2	2	no	—
416	C₂⁴	—	2	4	no	—
417	C₂⁴	—	2	4	no	—
418	C₂⁴	—	4	2	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	4	4	no	—
421	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
422	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
423	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
424	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
425	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
426	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
427	C₂×D₄	—	2	16	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	4	16	no	—
430	C₂×D₄	—	4	8	no	—
431	C₂×D₄	—	4	4	no	—
432	C₂×D₄	—	4	32	no	—
433	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
434	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
435	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
436	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	8	no	—
441	C₂²×C₄	—	4	16	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	8	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	—	4	8	no	—
451	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
452	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
453	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
454	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
455	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
456	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
457	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
466	C₄⋊C₄	—	2	8	no	—
467	C₄⋊C₄	—	4	4	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	—	4	2	no	—
470	C₄⋊C₄	—	4	8	no	—
471	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
472	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	4	4	no	—
478	C₂²⋊C₄	—	4	8	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	4	8	no	—
481	C₂²⋊C₄	—	4	8	no	—
482	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
483	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
484	C₄²	—	2	8	no	—
485	C₄²	—	4	8	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	1	no	—
491	C₂³	—	2	8	no	—
492	C₂³	—	2	8	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	8	no	—
497	C₂³	—	2	8	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	16	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	4	8	no	—
504	C₂³	—	4	8	no	—
505	C₂³	—	4	8	no	—
506	C₂³	—	4	8	no	—
507	C₂³	—	8	8	no	—
508	C₂³	—	8	8	no	—
509	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
510	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
511	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
512	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
513	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
514	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
515	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
516	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
517	D₄	—	2	32	no	—
518	D₄	—	2	16	no	—
519	D₄	—	4	8	no	—
520	D₄	—	4	16	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	8	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	8	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	8	8	no	—
542	C₂×C₄	—	8	16	no	—
543	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
544	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
545	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
546	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
547	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
548	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	4	no	—
552	C₂²	—	2	4	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	8	no	—
559	C₂²	—	2	8	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	8	no	—
563	C₂²	—	4	16	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	8	no	—
566	C₂²	—	4	8	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	8	16	no	—
569	C₂²	#518×C₆₇	1	4	yes	—
570	C₂²	#520×C₆₇	1	4	yes	—
571	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
572	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
573	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
574	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
575	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	8	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	8	no	—
583	C₄	#518×C₆₇	1	4	yes	—
584	C₂	—	2	4	no	—
585	C₂	—	2	4	no	—
586	C₂	—	4	2	no	—
587	C₂	—	4	8	no	—
588	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
589	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
590	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
591	C₂	C₆₇×#3536	1	2	yes	—
592	C₂	C₆₇×#3537	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2377	C₂	semidirect
3. C₆₇×#2361	C₂	semidirect
4. C₆₇×#2345	C₂	semidirect
5. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
6. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
7. C₆₇×#2325	C₂	semidirect
8. C₆₇×#2324	C₂	semidirect
9. C₆₇×#2315	C₂	semidirect
10. C₆₇×#2312	C₂	semidirect
11. C₆₇×#2280	C₂	semidirect
12. C₂×#380×C₆₇	C₂	semidirect
16. C₂×#121×C₆₇	C₄	semidirect
21. C₂×C₄×D₄×C₆₇	C₂²	semidirect
22. C₂×C₄×D₄×C₆₇	C₂²	semidirect
23. C₂×#117×C₆₇	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
29. #390×C₆₇	C₂²	semidirect
30. #390×C₆₇	C₂²	semidirect
31. #389×C₆₇	C₂²	semidirect
32. #388×C₆₇	C₂²	semidirect
33. #388×C₆₇	C₂²	semidirect
34. #387×C₆₇	C₂²	semidirect
35. #386×C₆₇	C₂²	semidirect
36. #386×C₆₇	C₂²	semidirect
37. #386×C₆₇	C₂²	semidirect
38. #385×C₆₇	C₂²	semidirect
39. #385×C₆₇	C₂²	semidirect
40. #381×C₆₇	C₂²	semidirect
46. #380×C₆₇	C₂²	semidirect
47. #380×C₆₇	C₂²	semidirect
48. #380×C₆₇	C₂²	semidirect
51. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
55. C₂×#95×C₆₇	C₂²	semidirect
60. C₂²×D₄×C₆₇	C₂×C₄	semidirect
61. C₂²×D₄×C₆₇	C₂×C₄	semidirect
67. C₂³×C₄×C₆₇	C₂×C₄	semidirect
77. #121×C₆₇	C₂×C₄	semidirect
78. #121×C₆₇	C₂×C₄	semidirect
79. #121×C₆₇	C₂×C₄	semidirect
80. #121×C₆₇	C₂×C₄	semidirect
85. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
95. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
287. #34379	C₆₇	direct (N × H)
298. #2377	C₂×C₆₇	semidirect
299. #2361	C₂×C₆₇	semidirect
300. #2345	C₂×C₆₇	semidirect
301. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
302. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
303. #2325	C₂×C₆₇	semidirect
304. #2324	C₂×C₆₇	semidirect
305. #2315	C₂×C₆₇	semidirect
306. #2312	C₂×C₆₇	semidirect
307. #2280	C₂×C₆₇	semidirect
308. C₂×#380	C₂×C₆₇	semidirect
309. C₆₇	#34379	direct (N × H)
313. C₂×#121	C₄×C₆₇	semidirect
318. C₂×C₄×D₄	C₂²×C₆₇	semidirect
319. C₂×C₄×D₄	C₂²×C₆₇	semidirect
320. C₂×#117	C₂²×C₆₇	semidirect
322. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
326. #390	C₂²×C₆₇	semidirect
327. #390	C₂²×C₆₇	semidirect
328. #389	C₂²×C₆₇	semidirect
329. #388	C₂²×C₆₇	semidirect
330. #388	C₂²×C₆₇	semidirect
331. #387	C₂²×C₆₇	semidirect
332. #386	C₂²×C₆₇	semidirect
333. #386	C₂²×C₆₇	semidirect
334. #386	C₂²×C₆₇	semidirect
335. #385	C₂²×C₆₇	semidirect
336. #385	C₂²×C₆₇	semidirect
337. #381	C₂²×C₆₇	semidirect
343. #380	C₂²×C₆₇	semidirect
344. #380	C₂²×C₆₇	semidirect
345. #380	C₂²×C₆₇	semidirect
348. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
351. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
352. C₂×#95	C₂²×C₆₇	semidirect
357. C₂²×D₄	C₂×C₄×C₆₇	semidirect
358. C₂²×D₄	C₂×C₄×C₆₇	semidirect
364. C₂³×C₄	C₂×C₄×C₆₇	semidirect
374. #121	C₂×C₄×C₆₇	semidirect
375. #121	C₂×C₄×C₆₇	semidirect
376. #121	C₂×C₄×C₆₇	semidirect
377. #121	C₂×C₄×C₆₇	semidirect
382. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
392. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#121×C₆₇	2
C₆₇×#2377	2
C₆₇×#2361	2
C₆₇×#2345	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#2325	2
C₆₇×#2324	2
C₆₇×#2315	2
C₆₇×#2312	2
C₆₇×#2280	2
C₂×#380×C₆₇	2
#34379	67

Subgroups — C67×#34379 #30713630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34379 #30713630